2o8 THE GRAMMAR OF SCIENCE 



us how to shift our position from O to P. Step so many 

 feet with such and such a bearing, and we shall pass from 

 O to P. 



The conception of bearing is so important that we must 

 say a few words more about it. The statement merely of 

 P's distance from O would carry us to any point whatever 

 on a sphere about O as centre. To fix a point on this 

 sphere we require the knowledge of at least two additional 

 independent points or elements. For example, a point 

 which we may term the " pole," Z, of the sphere would 

 serve for one. The opposite pole to Z would not serve 

 for the other, for it is not independent, but obtained by 

 producing ZO to cut the sphere again. Neither would 

 the " equator " corresponding to the polar line OZ serve 

 our purpose, for it again is not independent of OZ. But 

 a point X on this equator is independent of OZ and will 

 do very well. The plane through the lines OX and OZ 

 cuts the sphere in a " meridian," and if we take XOZ as 

 the meridian to help us determine " bearing," we may speak 

 of it as a prime meridian. If we take a line OX per- 

 pendicular to this prime meridian, it will cut the circle in 

 a point Y, and the system of lines OX, OY, OZ, each at 

 right-angles to the other two, is conveniently termed a 

 " frame of reference." There are many other ways of 

 determining bearing, but they can all be reduced to the 

 consideration of a frame of reference. Before, then, we 

 picture to ourselves any motion of a point P, we must 

 have selected an " origin of reference " O to give the 

 distance and a " frame of reference " OX, OY, OZ to give 

 the bearing. 



Thus if P be in motion and we know what is the step 

 from O to P at each instant of the motion, we shall have 

 a complete picture of the sequences of positions, the 

 motion of P relative to O and its frame. The reader 

 must be careful to notice the relativity of the motion ; 

 absolute motion, like absolute position, is inconceivable : 

 a point P is conceived as describing a path relatively to 

 something else. Thus the button on the man's waistcoat 

 moved relatively to the staircase which serves as frame, 



